Moran’s I

Moran’s I has been a form of measuring spatial correlation since the 1950, and it is likely the most widely adopted measure. It measures whether similar values cluster together (positive autocorrelation) or are more dispersed (negative autocorrelation). It is largely synonymous to the calculation of the correlation coefficient which measures the direction and strength between two variables where there their respective differences from the mean are multiplied and divided by the sum of squared differences from the mean. With Moran’s I, the product of the difference of means in the numerator is modified by a weight matrix that reflects the relevant observation to observation neighbor relationship. If a basic queen or rook neighbor relationship is used or a distance threshold smaller than the geographic extent of the data, the weight matrix is likely 0 for several observations not adjacent to each other. The specific equation for Moran’s I ($I$) is provided in Equation 1.

\[ \begin{aligned} (Eq. 1): I= &\frac{n}{\sum_{i=1}^n \sum_{j=1}^n w_{ij}}\frac{\sum_{i=1}^n \sum_{j=1}^n w_{ij}(X_i - \bar{X}) (X_j - \bar{X})}{\sum_{i=1}^n (X_i - \bar{X})^2} \end{aligned} \]

\[ \begin{aligned} I &= \text{Moran's I} \\ n &= \text{Number of observations, whether points or shapes} \\ w_{ij} &= \text{weight indexing location of i relative to j} \\ \bar{X} &= \text{Number of observations in the samplel} \\ X_i &= \text{Variable value at location i} \\ X_j &= \text{Variable value at location j} \\ \end{aligned} \]

For the purpose of this analysis, I am using a weight matrix associated with the neighbor definition of 25 miles, or 40,234 meters in the map projection of meters for WGS 84 EPSG:3857. In preliminary analysis, weight matrices were considered for 1 mile to 125 miles and 25 miles was selected as a result of considering spatial relationships that were coherent with the data. 25 miles was likely to encompass a city, whereas 125 miles didn’t have an apparent theoretical justification for two properties having a spatial relationship. The following map illustrates the neighbor relationships in they study area associated with 25 miles.

It is noted that 125 miles did show evidence of continuing spatial autocorrelation. Subsequent insight after completion of this paper’s analysis did indicate that (a) the R script was incorrectly specified during the initial examination of multiple weight matrices which created the illusion that 25 miles was significant and (b) it is generally a better practice to select the highest distance threshold where spatial autocorrelation is still present. Statisticians may also conduct the analysis with multiple weight matrices to avoid having current theoretical justifications determine the scope of the analysis and the results, avoiding confirmation bias. As an example, HUD provides direct oversight of public housing and conducts the REAC inspections in the data set, and has regional offices which may provide a theoretical motivation for significant neighbor relationships over 125 miles.

Hypothesis Testing

Moran’s I is an individual calculation, but is used in significant testing with the deployment of a random permutations of a Monte Carlo simulation. The observed variable is randomly assigned across the differently spaced spatial points, with repeated calculations of the Moran’s I (999 times in the case of this paper). This generates a particular random distribution that the observed Moran’s I can be measured against, resulting in ranked ordering of Moran’s I values that can be divided by the number of permutations to generate a pseudo p-value that can be measured against \(\alpha=5\%\). The null hypothesis is that there is no spatial autocorrelation and that the observed Moran’s I is within acceptable bounds of random variation. The alternative hypothesis is that there is autocorrelation and that observed Moran’s I is sufficiently unlikely if null hypothesis were true.

Local Indices of Spatial Autocorrelation (LISA)

The LISA procedure determines to what extent do sites in vicinity of a point \(i\) have values that are associated with the value observed at point \(i\). It creates a Moran’s I at every specific location, which is then plotted based on whether each point is positively or negatively spatially autocorrelated with its neighbors. When high values of the observation are associated with high values of it’s neighbors, it falls on the top-left quadrant of the LISA cluster map. When low values are associated with low values of neighbors, these points are plotted on the bottom-left quadrant and so forth. This quadrant classifications will be reflected on a map to illustrate the spatial relationship.

Hypothesis testing is done by doing many permutations in which all points except for the local value are shuffled, and the local Moran’s I is re-calculated. Similar to the global Moran’s I, these local calculations of Moran’s I are ranked for each point, with the rank order used to calculate a pseudo p-value that can be assessed using an threshold such as \(\alpha=5\%\). This p-value can also be mapped using the dataset’s geometry as a visual examination of where spatial autocorrelation is.

Because the weight matrix of 25 miles permits observations without neighbors, 58 observations were ignored in order to perform this LISA calculations in R.

Spatial Leg Model (SLM)

Spatial lag regression is a nested form of OLS regression that includes a spatially lagged variable for the independent variable, meaning the mean value observed at the neighboring observations for each observation subject to the specified weight matrix. This model assumes that the value of the dependent variable at one location is associated with the values of these nearby locations.

The regression model is demonstrated in Equation #2:

\[ \begin{aligned} (Eq. 2) hardcost\_per\_unit = & \rho * W y + \\ &\beta_0 + \beta_1 * inspection\_score + \beta_2 * FMR.2.BR + \\ &\beta_3 * log(Total.PIC.Units) + \beta_4 * PUM.Contract.Rent + \\ &\beta_5 * pct\_01br + \beta_6 * pct\_converted \\ &\beta_7 * pct\_black + \beta_8 * pct\_owners + \\ &\beta_9 * pct\_vacant + \beta_10 * urbanicity + \epsilon \end{aligned} \]

• \(\beta_0 \dots \beta_{10}\): Variable coefficients are the same as in Report #1 (with the exception of two new variables, PUM.Contract.Rent, pct_converted, and urbanicity) and are defined in the Appendix of this report.
• \(\rho\): This is similar to the coefficients on the OLS variables but reflects the amount of spatial autocorrelation in the dependent variables (ranging from perfect negative spatial autocorrelation of -1 to perfect positive spatial autocorrelation of 1).
• \(Wy\): This is the additional lagged predictor \(Wy\) added to the model, reflecting the (mean) value of the dependent variable’s neighbors.

Spatial Error Regression

Spatial error regression operates on the assumption that the residual at one location is associated with the residuals at neighboring locations. It performs a first-step regression typical of OLS, then regresses the residual term on the mean of the neighboring residuals. It takes the spatial information hat remains in the residual term and splits it up into a portion that is a spatial relationship (akin to calculating the Moran’s I) and a portion that is random noise. The coefficient generated is constrained between -1 and 1.

The spatial error regression model is demonstrated in Equation #3:

\[ \begin{aligned} (Eq. 3) hardcost\_per\_unit = &\beta_0 + \beta_1 * inspection\_score + \beta_2 * FMR.2.BR + \\ &\beta_3 * log(Total.PIC.Units) + \beta_4 * PUM.Contract.Rent + \\ &\beta_5 * pct\_01br + \beta_6 * pct\_converted \\ &\beta_7 * pct\_black + \beta_8 * pct\_owners + \\ &\beta_9 * pct\_vacant + \beta_10 * urbanicity + \\ & \lambda * W \epsilon + u \\ \end{aligned} \]

• \(\beta_0 \dots \beta_{10}\): Coefficients on main predictors have the same interpretation as the spatial lag model.
• \(\epsilon\): The residual (unobserved variation in the data) in the first-stage OLS model, which also serves as the independent variable of the second-stage regression.
• \(\lambda\): The observed spatial autocorrelation of the residuals of the model, i.e., how much residuals in one location are correlated with the residuals of their neighbors. It is bounded by -1 (perfect negative spatial autocorrelation) and 1 (perfect positive autocorrelation).
• \(W \epsilon\): The (mean) residual of the neighboring points to the independent variable of the second-stage model.
• \(u\): The residual of the second-stage model, representing random noise in the model once autocorrelation has been accounted for.

Model Diagnostics

All assumptions identified for OLD are still needed in spatial lag and spatial error regression with the exception of spatial independence of observations. The goal of these two models is to account for spatial autocorrelation, but they do so in a different fashion. The spatial lag model incorporates spatial structure with neighbors in the dependent variable predicted by the model, useful in instances where observation impact nearby ones (in the case of externalities of the dependent variable). The spatial error model, in contrast, signals the presence of unobserved (spatial) predictors accounting for a portion of the residual. In either case, a significant \(\rho\) or \(\lambda\) coefficient indicates the presence of spatial autocorrelation.

I will compare the results of both the spatial lag and spatial error regression models with OLS to decide which spatial models are an improvement. There are three criteria that will be used:

• Akaike Information Criterion (AIC)/Schwarz Criterion (SC): AIC and SC measure the goodness of fit of an estimated model, assessing how much information is lost when a model seeks to capture reality. They balance the tradeoff between a more precise model, and one that is more complex. Models with lower scores on these criteria fit better.
  
• Log Likelihood: This measure relies on the maximum likelihood approach to estimating parameters in which coefficients are chosen such that the observed data has the greatest chance of occurring. It is a nested model and, thus, can only be used to compare spatial lag model or spatial error model to OLS, not to each other. In other words, it determines whether the additional predictors (the spatial predictors) reflect a significant improvement over OLS in fitting the data in our dataset. This measure is always negative (it’s the log of a percentage) and when the spatial model is higher (less negative) it is considered an improvement.
• Likelihood Ratio Test: This measure converts Log Likelihood into a hypothesis test where the null hypothesis is that one of the spatial models is not better than OLS (and we should not continue use of them in this instance) and the alternative hypothesis is that they are better specifications for the observed data. When the p-value is lower than a specified threshold (typically \(\alpha=5\%\)) than we reject the null.

Another way of comparing spatial lag and error results to OLS is by calculating the Moran’s I of their regression residuals. The introduction of either model should lessen the spatial autocorrelation when compared to the original OLS. The model which has less spatial autocorrelation (i.e. a Moran’s I closer to 0) would more effective in accomplishing teh goal. If one or both models does not significantly reduce the Moran’s I of the residual, they are either not capturing the nature of the of spatial relationship or the weight matrix may need to be adjusted.

Table 1: Ordinary Least Squares Regression Results**

## 
## Call:
## lm(formula = hardcost_per_unit ~ latest_score + lTotal.PIC.Units + 
##     PUM.Contract.Rent + FMR.2.BR + pct_01br + pct_converted + 
##     pct_black + pct_owners + pct_vacant + urbanicity, data = rad_south_sf_3857)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -101441  -26846  -12350   18137  227349 
## 
## Coefficients:
##                    Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)        46614.64   26594.57   1.753              0.08021 .  
## latest_score        -508.21     156.60  -3.245              0.00125 ** 
## lTotal.PIC.Units    6940.65    2646.21   2.623              0.00897 ** 
## PUM.Contract.Rent     68.45      26.85   2.549              0.01107 *  
## FMR.2.BR             -13.17      14.38  -0.916              0.35992    
## pct_01br            8803.39    6836.67   1.288              0.19842    
## pct_converted     -59740.06    6753.49  -8.846 < 0.0000000000000002 ***
## pct_black          18735.31    7402.23   2.531              0.01166 *  
## pct_owners          4642.17   12055.32   0.385              0.70034    
## pct_vacant         16692.01   25661.53   0.650              0.51567    
## urbanicity         10452.45    5506.01   1.898              0.05819 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 44470 on 533 degrees of freedom
## Multiple R-squared:  0.2534, Adjusted R-squared:  0.2393 
## F-statistic: 18.09 on 10 and 533 DF,  p-value: < 0.00000000000000022

## [1] "Results of the Log Likelihood Calculation:"

## 'log Lik.' -6588.56 (df=12)

There are five variables are statistically significant at \(\alpha=5\%\): the latest inspection score of the public housing site, the number of public housing units at the site, the initial subsidy offered to converting sites, the percentage of pre-program units that have been converted and the black percentage of the census tract. Four variables are not close to being significant: the FMR for a 2BR unit, the percentage of 0-1BR units, the homeownership rate and the percentage of vacant housing in the census tract. Urbanicity is close to being significant, but not given the \(a\) level. The \(R^2\) of 0.2534 indicates that 25% of the variation in the hard cost spending per unit is accounted by changes in these 10 predictors. Report #1 dealt with data from a larger study area, and didn’t adequately account for how destination RAD transactions mix and match sending public housing sites. However, the current model reflects a slight improvement in overall explanation of changes in the dependent variable.

The distribution and standardized residual plot are provided for the OLS model below. They indicate normality, but a clear pattern in relationship to the predicted values of the dependent variable.

Base OLS Model Plots

In addition to a visual examination, the Breusch-Pagan Test, Koenker-Bassett Test (Studentized Breusch-Pagan Test) and the White Test all show p-values below \(\alpha=5\%\) which lead to rejection of the null hypothesis of homoskedasticity. This supports the appearance of heteroskedasticity in the plots.

## 
##  Breusch-Pagan test
## 
## data:  reg
## BP = 59.202, df = 10, p-value = 0.00000000513

## 
##  studentized Breusch-Pagan test
## 
## data:  reg
## BP = 22.82, df = 10, p-value = 0.01143

## White's test results
## 
## Null hypothesis: Homoskedasticity of the residuals
## Alternative hypothesis: Heteroskedasticity of the residuals
## Test Statistic: 89.99
## P-value: 0

The Jarque-Bera test shows that the null hypothesis of normally distributed residuals of the OLS regression is also rejected in favor of non-normality. This demonstrates how a visual examination is often inadequate because the histogram of the residuals is not obviously non-normal.

## 
##  Jarque Bera Test
## 
## data:  reg$residuals
## X-squared = 406.54, df = 2, p-value < 0.00000000000000022

A map of the OLS standardized residuals is included, followed by a scatter plot of the residuals against their lagged neighbors and the group of Moran’s I calculations and Monte Carlo significance test. The map doesn’t immediately show a pattern as more extreme residuals are surrounded by less extreme residuals. The scattered plot of the lagged residuals has a positive slope, with a Moran’s I of .19 that is significant against 999 permutations. This indicates significant spatial autocorrelation at the weight matrix of 25 miles. This is suggests that nearby public housing sites with RAD conversions see more similar hard cost spending in associated transactions than two random sites selected from the data set, and that OLS procedures will be unreliable.

## 
##  Monte-Carlo simulation of Moran I
## 
## data:  x 
## weights: wfile  
## number of simulations + 1: 1000 
## 
## statistic = 0.19221, observed rank = 1000, p-value <
## 0.00000000000000022
## alternative hypothesis: two.sided

Spatial Lag Regression

Table 2 below provides the results of the Spatial Lag Model (SLM).

## 
## Call:lagsarlm(formula = hardcost_per_unit ~ latest_score + lTotal.PIC.Units + 
##     PUM.Contract.Rent + FMR.2.BR + pct_01br + pct_converted + 
##     pct_black + pct_owners + pct_vacant + urbanicity, data = rad_south_sf_3857, 
##     listw = weight_matrix_25, zero.policy = TRUE)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -119416.3  -25118.2   -9025.6   14788.4  239594.6 
## 
## Type: lag 
## Regions with no neighbours included:
##  132 214 216 291 306 318 323 344 356 374 376 382 390 399 401 920 929 936 950 966 1058 1189 1200 1209 1334 1335 1351 1966 2038 2127 2234 2243 3044 3053 3083 3084 3092 3216 3337 3362 5216 5225 5235 5246 5254 5255 5270 5408 5451 5452 5492 5519 5638 5727 5737 5778 5788 5930 
## Coefficients: (numerical Hessian approximate standard errors) 
##                     Estimate Std. Error z value              Pr(>|z|)
## (Intercept)        35596.363  24613.524  1.4462              0.148118
## latest_score        -487.448    146.262 -3.3327              0.000860
## lTotal.PIC.Units    6197.466   2465.436  2.5137              0.011946
## PUM.Contract.Rent     71.232     24.993  2.8501              0.004371
## FMR.2.BR             -11.523     13.392 -0.8604              0.389580
## pct_01br            5965.182   6261.856  0.9526              0.340782
## pct_converted     -54027.701   6363.352 -8.4904 < 0.00000000000000022
## pct_black          11557.224   7112.121  1.6250              0.104162
## pct_owners          5363.998  11818.068  0.4539              0.649914
## pct_vacant         27781.717  24166.401  1.1496              0.250308
## urbanicity          6057.606   5149.243  1.1764              0.239432
## 
## Rho: 0.31789, LR test value: 47.41, p-value: 0.0000000000057586
## Approximate (numerical Hessian) standard error: 0.043411
##     z-value: 7.3227, p-value: 0.00000000000024292
## Wald statistic: 53.622, p-value: 0.00000000000024303
## 
## Log likelihood: -6564.855 for lag model
## ML residual variance (sigma squared): 1729500000, (sigma: 41587)
## Number of observations: 544 
## Number of parameters estimated: 13 
## AIC: 13156, (AIC for lm: 13201)

## 
##  Breusch-Pagan test
## 
## data:  
## BP = 78.099, df = 10, p-value = 0.000000000001183

## 
##  studentized Breusch-Pagan test
## 
## data:  
## BP = 24.526, df = 10, p-value = 0.00632

## 
##  Likelihood ratio for spatial linear models
## 
## data:  
## Likelihood ratio = 47.41, df = 1, p-value = 0.000000000005759
## sample estimates:
## Log likelihood of lagreg25      Log likelihood of reg 
##                  -6564.855                  -6588.560

## 
##  Monte-Carlo simulation of Moran I
## 
## data:  x 
## weights: wfile  
## number of simulations + 1: 1000 
## 
## statistic = 0.0014521, observed rank = 590, p-value = 0.82
## alternative hypothesis: two.sided

Spatial Error Regression

Table 2 below provides the results of the Spatial Error Model (SEM).

## 
## Call:
## errorsarlm(formula = hardcost_per_unit ~ latest_score + lTotal.PIC.Units + 
##     PUM.Contract.Rent + FMR.2.BR + pct_01br + pct_converted + 
##     pct_black + pct_owners + pct_vacant + urbanicity, data = rad_south_sf_3857, 
##     listw = weight_matrix_25, zero.policy = TRUE)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -113681.0  -25149.0   -9963.4   15888.6  229424.9 
## 
## Type: error 
## Regions with no neighbours included:
##  132 214 216 291 306 318 323 344 356 374 376 382 390 399 401 920 929 936 950 966 1058 1189 1200 1209 1334 1335 1351 1966 2038 2127 2234 2243 3044 3053 3083 3084 3092 3216 3337 3362 5216 5225 5235 5246 5254 5255 5270 5408 5451 5452 5492 5519 5638 5727 5737 5778 5788 5930 
## Coefficients: (asymptotic standard errors) 
##                      Estimate  Std. Error z value           Pr(>|z|)
## (Intercept)        34246.2677  28128.6016  1.2175           0.223418
## latest_score        -570.2427    157.6534 -3.6171           0.000298
## lTotal.PIC.Units    6505.2698   2655.2363  2.4500           0.014287
## PUM.Contract.Rent     77.2892     26.5962  2.9060           0.003661
## FMR.2.BR              -6.6643     17.9105 -0.3721           0.709828
## pct_01br            3959.3009   6418.0241  0.6169           0.537298
## pct_converted     -44894.2830   6143.1236 -7.3081 0.0000000000002711
## pct_black          14501.3478   8071.1301  1.7967           0.072384
## pct_owners          4580.1290  11948.3105  0.3833           0.701476
## pct_vacant         21176.3707  24934.8459  0.8493           0.395732
## urbanicity         10526.9254   6356.6005  1.6561           0.097709
## 
## Lambda: 0.35109, LR test value: 40.116, p-value: 0.00000000023937
## Approximate (numerical Hessian) standard error: 0.050661
##     z-value: 6.9303, p-value: 0.0000000000042009
## Wald statistic: 48.028, p-value: 0.0000000000042009
## 
## Log likelihood: -6568.502 for error model
## ML residual variance (sigma squared): 1741600000, (sigma: 41732)
## Number of observations: 544 
## Number of parameters estimated: 13 
## AIC: 13163, (AIC for lm: 13201)

## 
##  Breusch-Pagan test
## 
## data:  
## BP = 92.241, df = 10, p-value = 0.000000000000001887

## 
##  studentized Breusch-Pagan test
## 
## data:  
## BP = 32.283, df = 10, p-value = 0.0003592

## 
##  Likelihood ratio for spatial linear models
## 
## data:  
## Likelihood ratio = 40.116, df = 1, p-value = 0.0000000002394
## sample estimates:
## Log likelihood of errreg    Log likelihood of reg 
##                -6568.502                -6588.560

## 
##  Monte-Carlo simulation of Moran I
## 
## data:  x 
## weights: wfile  
## number of simulations + 1: 1000 
## 
## statistic = -0.0043904, observed rank = 503, p-value = 0.994
## alternative hypothesis: two.sided

Table 4: Geographically Weighted Regression Results

## Call:
## gwr(formula = hardcost_per_unit ~ latest_score + lTotal.PIC.Units + 
##     PUM.Contract.Rent + FMR.2.BR + pct_01br + pct_converted + 
##     pct_black + pct_owners + pct_vacant + urbanicity, data = rad_south_sf_3857s, 
##     gweight = gwr.Gauss, adapt = bw, hatmatrix = TRUE, se.fit = TRUE)
## Kernel function: gwr.Gauss 
## Adaptive quantile: 0.117666 (about 64 of 544 data points)
## Summary of GWR coefficient estimates at data points:
##                           Min.      1st Qu.       Median      3rd Qu.
## X.Intercept.      -80800.80636  -7597.10929  27664.35889  46711.00002
## latest_score        -933.98712   -670.97199   -554.30474   -489.31860
## lTotal.PIC.Units  -10787.67899   1520.48056   4935.69979   9527.29268
## PUM.Contract.Rent     32.20415     56.20936     87.78761    103.98364
## FMR.2.BR             -84.13565     -0.57498     44.45732    102.65491
## pct_01br          -11932.10688   3894.34891   9605.10537  12306.00275
## pct_converted     -80586.60548 -70283.89796 -59836.18604 -47583.49816
## pct_black         -20242.12552   1123.91940   7373.36548  26258.97156
## pct_owners        -31426.70926 -19041.81877  -9911.69325   3743.99090
## pct_vacant        -17557.54230  -1092.04033  12125.72073  31170.48102
## urbanicity         -9003.57037  -2434.58818   1718.63991   7913.40250
##                           Max.     Global
## X.Intercept.      130484.97441  46614.638
## latest_score        -288.16600   -508.214
## lTotal.PIC.Units   15842.05314   6940.654
## PUM.Contract.Rent    174.94389     68.454
## FMR.2.BR             138.88833    -13.174
## pct_01br           16835.55620   8803.390
## pct_converted     -28469.99977 -59740.056
## pct_black          51863.39091  18735.305
## pct_owners         58005.15362   4642.169
## pct_vacant        190953.11716  16692.013
## urbanicity         16807.58788  10452.453
## Number of data points: 544 
## Effective number of parameters (residual: 2traceS - traceS'S): 68.98078 
## Effective degrees of freedom (residual: 2traceS - traceS'S): 475.0192 
## Sigma (residual: 2traceS - traceS'S): 41762.7 
## Effective number of parameters (model: traceS): 51.20744 
## Effective degrees of freedom (model: traceS): 492.7926 
## Sigma (model: traceS): 41002.66 
## Sigma (ML): 39025.16 
## AICc (GWR p. 61, eq 2.33; p. 96, eq. 4.21): 13161.83 
## AIC (GWR p. 96, eq. 4.22): 13097.31 
## Residual sum of squares: 828491811147 
## Quasi-global R2: 0.4131648

# Assuming you have the design matrix X
X <- model.matrix(~ latest_score +
                lTotal.PIC.Units +
                PUM.Contract.Rent +
                FMR.2.BR +
                pct_01br +
                pct_converted +
                pct_black +
                pct_owners +
                pct_vacant +
                urbanicity,
                data=rad_south_sf_3857s)

# Perform Singular Value Decomposition (SVD)
svd_result <- svd(X)

# Extract the singular values
singular_values <- svd_result$d

# Calculate the condition number
condition_number <- max(singular_values) / min(singular_values)

# Print the condition number
print(condition_number)

## [1] 14589.91

The table compares AIC and Moran’s I across all 4 models, and is followed by the group of Moran’s I plots and significant test for the standardized residuals. GWR fits the better data, but doesn’t appear to remedy all spatial autocorrelation even when it expands the general neighborhood of each observation.

## 
##  Monte-Carlo simulation of Moran I
## 
## data:  x 
## weights: wfile  
## number of simulations + 1: 1000 
## 
## statistic = 0.13246, observed rank = 1000, p-value <
## 0.00000000000000022
## alternative hypothesis: two.sided

The following map shows the standardized residuals across the study area. There is a cluster of extreme values in Atlanta and Nashville, but no otherwise apparent patterns.

Table 4 examines the local GWR coefficients in contrast to the original OLS estimates under the “Global”. There is significant variation in the intercept, but it was and remains insignificant. Beyond that, there is noticeable local variation in the coefficient for the log transformation of the number of units.

The resulting map of the standardized shows the spatial location of the coefficients.

• latest_score: Lower inspection scores ae consistently leading to higher per-unit spending, with a lower impact in western North Carolina.
• lTotal.PIC.Units: In North. Carolina, increases in the size of public housing sites lead to less funding (but in a decreasing rate). Everywhere else, they lead to more funding.
• PUM.Contract.Rent: The Southeastern part of the map experiences a higher impact of increased contract rents in the amount of investment received by a property.
• pct_01br: Smaller bedroom units are more of a negative factor in Tennessee and Northern Alabama and Mississippi.
• pct_converted: The percentage converted is a significant negative factor everywhere but less in Western North Carolina.
• pct_black & pct_owners: The black percentage and ownership rates of the census tract is most significant at the nexus of Texas, Louisiana and Arkansas.
• pct_vacant: Vacant housing rates are most significant in North Carolina and Virginia.
• Urbanicity: Urbanicity is most significant along the border of Texas and Mexico.

The final maps presents the local \(R^2\) calculations that range from 0.2 to 0.3 (Atlanta, Charlotte, Knoxville, Nashville) to 0.5 to 0.6 (Mississippi). This reflects where the 10 predictors do a better job of accounting for variation.